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Popular Trigonometry >

30=15cos((2pi)/(365)t)+43

Solution

30 = 15 cos (\frac{2 π}{365} t) + 43

Solution

t = \frac{365 \cdot 2.61927 \dots}{2 π} + 365 n, t = - \frac{365 \cdot 2.61927 \dots}{2 π} + 365 n

Degrees

t = 8718.00664 \dots^{\circ} + 20912.95952 \dots^{\circ} n, t = - 8718.00664 \dots^{\circ} + 20912.95952 \dots^{\circ} n

Solution steps

30 = 15 cos (\frac{2 π}{365} t) + 43

Switch sides

15 cos (\frac{2 π}{365} t) + 43 = 30

Move

43

to the right side

15 cos (\frac{2 π}{365} t) + 43 = 30

Subtract

43

from both sides

15 cos (\frac{2 π}{365} t) + 43 - 43 = 30 - 43

Simplify

15 cos (\frac{2 π}{365} t) = - 13

15 cos (\frac{2 π}{365} t) = - 13

Divide both sides by

15

15 cos (\frac{2 π}{365} t) = - 13

Divide both sides by

15

\frac{15 cos ( \frac{2 π}{365} t )}{15} = \frac{- 13}{15}

Simplify

cos (\frac{2 π}{365} t) = - \frac{13}{15}

cos (\frac{2 π}{365} t) = - \frac{13}{15}

Apply trig inverse properties

cos (\frac{2 π}{365} t) = - \frac{13}{15}

General solutions for

cos (\frac{2 π}{365} t) = - \frac{13}{15}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

\frac{2 π}{365} t = arccos (- \frac{13}{15}) + 2 πn, \frac{2 π}{365} t = - arccos (- \frac{13}{15}) + 2 πn

\frac{2 π}{365} t = arccos (- \frac{13}{15}) + 2 πn, \frac{2 π}{365} t = - arccos (- \frac{13}{15}) + 2 πn

Solve

\frac{2 π}{365} t = arccos (- \frac{13}{15}) + 2 πn : t = \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

\frac{2 π}{365} t = arccos (- \frac{13}{15}) + 2 πn

Multiply both sides by

365

\frac{2 π}{365} t = arccos (- \frac{13}{15}) + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} t = 365 arccos (- \frac{13}{15}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} t = 365 arccos (- \frac{13}{15}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} t : 2 π t

365 \cdot \frac{2 π}{365} t

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 \cdot 365 π}{365} t

Cancel the common factor:

365

= t \cdot 2 π

Simplify

365 arccos (- \frac{13}{15}) + 365 \cdot 2 πn : 365 arccos (- \frac{13}{15}) + 730 πn

365 arccos (- \frac{13}{15}) + 365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= 365 arccos (- \frac{13}{15}) + 730 πn

2 π t = 365 arccos (- \frac{13}{15}) + 730 πn

2 π t = 365 arccos (- \frac{13}{15}) + 730 πn

2 π t = 365 arccos (- \frac{13}{15}) + 730 πn

Divide both sides by

2 π

2 π t = 365 arccos (- \frac{13}{15}) + 730 πn

Divide both sides by

2 π

\frac{2 π t}{2 π} = \frac{365 arccos ( - \frac{13}{15} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π t}{2 π} = \frac{365 arccos ( - \frac{13}{15} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π t}{2 π} : t

\frac{2 π t}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{365 arccos ( - \frac{13}{15} )}{2 π} + \frac{730 πn}{2 π} : \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

\frac{365 arccos ( - \frac{13}{15} )}{2 π} + \frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Divide the numbers:

\frac{730}{2} = 365

= \frac{365 πn}{π}

Cancel the common factor:

π

= 365 n

= 365 n

= \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

t = \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

t = \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

t = \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

Solve

\frac{2 π}{365} t = - arccos (- \frac{13}{15}) + 2 πn : t = - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

\frac{2 π}{365} t = - arccos (- \frac{13}{15}) + 2 πn

Multiply both sides by

365

\frac{2 π}{365} t = - arccos (- \frac{13}{15}) + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} t = - 365 arccos (- \frac{13}{15}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} t = - 365 arccos (- \frac{13}{15}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} t : 2 π t

365 \cdot \frac{2 π}{365} t

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 \cdot 365 π}{365} t

Cancel the common factor:

365

= t \cdot 2 π

Simplify

- 365 arccos (- \frac{13}{15}) + 365 \cdot 2 πn : - 365 arccos (- \frac{13}{15}) + 730 πn

- 365 arccos (- \frac{13}{15}) + 365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= - 365 arccos (- \frac{13}{15}) + 730 πn

2 π t = - 365 arccos (- \frac{13}{15}) + 730 πn

2 π t = - 365 arccos (- \frac{13}{15}) + 730 πn

2 π t = - 365 arccos (- \frac{13}{15}) + 730 πn

Divide both sides by

2 π

2 π t = - 365 arccos (- \frac{13}{15}) + 730 πn

Divide both sides by

2 π

\frac{2 π t}{2 π} = - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π t}{2 π} = - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π t}{2 π} : t

\frac{2 π t}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

- \frac{365 arccos ( - \frac{13}{15} )}{2 π} + \frac{730 πn}{2 π} : - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

- \frac{365 arccos ( - \frac{13}{15} )}{2 π} + \frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Divide the numbers:

\frac{730}{2} = 365

= \frac{365 πn}{π}

Cancel the common factor:

π

= 365 n

= 365 n

= - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

t = - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

t = - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

t = - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

t = \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n, t = - \frac{365 arccos ( - \frac{13}{15} )}{2 π} + 365 n

Show solutions in decimal form

t = \frac{365 \cdot 2.61927 \dots}{2 π} + 365 n, t = - \frac{365 \cdot 2.61927 \dots}{2 π} + 365 n

Graph

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